4. Given the graph below; a) Determine if the graph below has an Eulerian cycle using theory rather than trial-and- error. Thoroughly justify your answer. If an Eulerian cycle exists, give a vertex list starting at vertex A. If it does not exist, is it possible to add one edge to the graph so that an Eulerian cycle now exists? If so, draw the edge on a copy of the graph, then find an Eulerian cycle starting at vertex A. If it is still not possible, justify your answer. b) Determine if the graph below has an Eulerian path using theory rather than trial-and- error. Thoroughly justify your answer. If it does exist, give a vertex list starting at the vertex of your choice. If it does not exist, is it possible to add one edge to the graph so that an Eulerian path now exists? If so, draw the edge on a copy of the graph, then find the Eulerian path starting at a vertex of your choice. If it is still not possible, justify your answer. c) Find a Hamiltonian cycle starting at vertex A. Draw the Hamiltonian cycle on the graph and list the vertices of the cycle. A B E H F G D Note: A Hamiltonian cycle is a simple cycle that traverses all vertices. A simple cycle starts at a vertex, visits other vertices once then returns to the starting vertex.